Pergamon
PII: S0360-5442(98)00021-8
Energy Vol. 23, No. 9, pp. 699–709, 1998

1998 Elsevier Science Ltd. All rights reserved
Printed in Great Britain
0360-5442/98 $19.00
+
0.00
I. G. BRYDEN, †‡ S. NAIK, † P. FRAENKEL § and C. R. BULLEN
얏
† The Robert Gordon University, School of Mechanical and Offshore Engineering, Schoolhill, Aberdeen,
UK; § IT Power Ltd, The Warren, Bramshill Road, Eversley, Hampshire, UK; 얏 Heriot-Watt University, The
International Centre for Island Technology, Department of Civil and Offshore Engineering, Orkney, UK
(Received 14 October 1997)
Abstract—This paper is an outline of a methodology which may be used to optimise the development of a tidal current resource. The size and potential of the European tidal current energy resource is indicated and a brief description of a potential technology option is given. The overlying principles of resource conversion are discussed and a methodology required to achieve optimal exploitation is presented. Aspects of the methodology are outlined using examples drawn from tidal current studies in the European Union (EU).

1998 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION AND THE EUROPEAN RESOURCE
Tidal currents represent a large and untapped energy resource. It has been estimated in a recent report for the European Commission Directorate General for Energy [1] that the European resource could represent a potential for 12,500 MW installed capacity. If even a small fraction of this potential were exploited, it could represent a major contribution to the European energy market and, as such, result in appreciable reductions in CO
2 emissions.
UK waters offer particularly attractive potential sources of tidal current power. A study funded by the UK Department of Trade and Industry [2] concluded that tidal currents could supply a major portion of the UK’s electricity requirements but that the cost could be prohibitive. At a regional scale, it has been suggested that electricity could be generated in the Northern Isles of Scotland for less than
5 p/kWhr [3,4]. No one doubts the size of the potential tidal current resource in Europe but there has not yet been a large-scale prototype study.
2. GENERATION OF ELECTRICITY
2.1. Technology options
Most recent studies have concentrated on horizontal axis turbine designs such as the one shown in
Fig. 1. This is mounted on a monopile seated in a drilled hole. Alternative suggestions have included floating support vessels and vertical axis turbines. It is anticipated that any device used in European waters in the near future will use a horizontal axis approach. Axial flow turbines have the advantage of being able to be yawed edge onto the flow in order to disable them should a fault develop. They also have more design flexibility and more compact geometries than any proposed vertical axis designs.
2.2. Conversion principles
A
0
Moving water carries kinetic energy. The energy per second intercepted by a device of frontal area
(m 2 ) in a current of speed U (m/s) is given by
P(t)
=
(1/2)
␳
A
0
U 3 (t), (1)
‡ Author for correspondence. Fax: (0)1224 262 333; e-mail: i.bryden@rgu.ac.uk
699

700 I. G. Bryden et al
Fig. 1. Artist’s impression of a horizontal axis tidal current turbine.
where
␳ is the water density (kg/m 3 ). The power that can be converted to a useable mechanical form is limited for a device in an open water flow to
P
T
(t)
=
(1/2)C p
␳
A
0
U 3 (t), (2) where C p is the power coefficient. The value of C p for a turbine in a flow of incompressible fluid is limited to a maximum theoretical value around 60%. The value of C p for a real device is generally a function of the ratio between the speed of the turbine tip and the flow speed, which is commonly known as the tip speed ratio. Fig. 2 shows a representative relationship between the tip speed ratio (
␭
) and the power coefficient (C p
). The actual shape of this curve is dependent upon the blade form and the
Fig. 2. Sample C p
-
␭ curve for a four- bladed turbine [4].

Matching tidal current plants to local flow conditions 701
Fig. 3. Power output curve assuming a 10-s rotational period (based upon the C p
-
␭ curve from Fig. 2).
number of blades. Just what should be deemed as “independent design variables” in a system specification are subject to investigation but, in any case, should include the shape of this curve. If a manufacturer could offer a range of turbine forms, each with a different C p
-
␭ curve, then the selection would be within the design specification. Most suggested designs for tidal turbines operate at a constant rotational speed. This allows a relationship between the flow speed and the power output to be determined. Fig. 3 displays the anticipated relationship between current Speed and Power output for a 20 m diameter turbine rotating once every 10 s, if the turbine characteristics are as shown in Fig. 2. The effect of, for example, speed of rotation, on the power curve can be substantial. If the period is increased to 12 s, the curve changes to that shown in Fig. 4.
3. FACTORS AFFECTING GENERATION COSTS
3.1. Geographic factors
3.1.1. Water depth.
The available depth of water has an obvious influence on the possible turbine size. It has been suggested [1] that the maximum turbine size is related to water depth, as shown in
Table 1. In the case where there is no exclusion of shipping, it is assumed that the top of the rotor needs to be at the lowest astronomic tide (LAT) minus 1.5 m for the lowest negative storm surge, minus 2.5 m for the trough of a 5 m wave and minus a further 5 m to minimise the potential for damage from shipping and waves. In addition, the bottom tip of the blades must not be within 25% of the water depth at LAT from the sea bed. If shipping is excluded from the vicinity of a turbine, the rules
Fig. 4. Power output curve assuming a 12-s rotational period (based upon the C p
-
␭ curve from Fig. 2).

702
Water depth
⬍
20 m
20–25 m
25–40 m
⬎
40 m
I. G. Bryden et al
Table 1. Influence of water depth on maximum permitted turbine size.
Rotor diameter (assuming no shipping
(exclusion)
Rotor diameter (assuming shipping exclusion)
5 m
10 m
20 m
10 m
120 m
20 m
20 m can be relaxed. An alternative rule in this case could be simply that the turbine diameter is 50% of the water depth and the hub should be at the midwater point.
3.1.2. Cyclic currents and parametric descriptions.
Tidal currents are not, of course, constant. They vary according to cycles governed by the motion of the Earth and the Moon. It is generally possible to parameterise tidal currents as series of simple sinusoids. If it is assumed that only the semidiurnal and spring–neap cycles need be considered, then the tidal currents can be simplified to the form
U x
(t)
=
A + [B + C cos(2
␲
t/T
1
)] cos(2
␲
t/T
0
), U y
(t)
=
F + [D + E cos(2
␲
t/T
1
)] sin(2
␲
t/T
0
), (3) where A and F are related to residial current speeds, B, C, D and E are amplitude terms, T
0 period of the semidiurnal variation, T
1 is the period of the spring–neap cycle, U x is the
(t) represents the E–
W current speed and U y
(t) represents the N–S current speed. In reality, a more complex form of parametric description may be required.
If, as in Fig. 1, the turbine can yaw, then it is the absolute value
兩
U(t)
兩 of the current speed which is important, where
兩
U(t)
兩 = √
U 2 x
(t) + U 2 y
(t).
(4)
In many locations, it will not be possible to parameterise the tidal currents in the simple form shown here but alternative parametric descriptions based upon the underlying astronomic mechanisms will always be available.
The influence of rotational speed upon the maximum and average power generated during the spring– neap cycle, assuming a 20 m diameter device with a C p
-
␭ curve as in Fig. 2, might be as shown in
Fig. 5. The situation is further complicated by the fact that the current speed varies through the water column. If the current follows the frequently assumed 1/7th power law, then
U(z)
=
0.93
×
(z/0.32h)
1
7
U peak
, (5)
Fig. 5. Influence of rotational speed on the peak and mean power outputs, for a turbine with the C p
-
␭ curve shown in Fig. 2 and the following current parameters: spring peak (North–South)
=
3.0 (m/s); neap peak
(North–South)
=
2.0 (m/s); spring peak (East–West)
=
1.0 (m/s); neap peak (East–West)
=
0.5 (m/s).

Matching tidal current plants to local flow conditions 703 where z
= distance above the sea bed, h
= water depth and U peak is the current speed at the surface and the flow might be expected to be as shown in Fig. 6. The 1/7th power law may be too simplistic for many locations. Estuarial environments in particular will require a more complex parametric description.
It is reasonable to assume that the current speed should be averaged over the swept area of the turbine. If the current profile can be specified in terms of a power law, then the value of the power factor, 1/7 in the case shown, becomes another parameter required to describe the flow conditions.
Indeed, it is conceivable that the value of the power factor might change through the tidal cycle. As a first approximation, however, a single power law coefficient might suffice. If the bulb of the turbine is at a height Z hub and the diameter is D, the average current speed over the swept area will be u¯
=
(1/
␲ r 2 )
+ r 冕
− r cos[(sin
−
1 (y/r))] u(y + z
0
)dy, (6) where r is the turbine radius, z
0 a distance
␰ above the sea bed.
is the height of the hub above the sea bed and u(
␰
) is the flow speed
Within a tidal channel, the flow conditions will be dependent upon the location within the channel.
If a sufficiently thorough description of the conditions is available it will be possible to identify a function which enables determination of the flow parameters as a function of location within the channel.
This will probably involve use of a look-up table. Fig. 7 shows the output from a computer model of the flow, TIDESIM, in a tidal channel. The diagram shows values of mean current speed as a function of position but the output could equally have delivered a full parametric description of the flow, including water depth. Use of such a model would allow the effective creation of position-dependent functions giving flow parameters described by
(P flow
) i
=
Function(x,y); (i
=
1,N), (7) where x, y are the E–W location and the N–S location and N is the required number of flow parameters.
3.1.3. Turbine location.
The location of a turbine within a channel will have an obvious implication upon costs, even without consideration of the nature of cable landfall. Cable length is a major proportion of overall costs. In particular, the distance from the turbine and the landfall must be considered. The cost of the cable will also be related to the required power rating. The cable cost must take into account pre-installation surveys, installation costs and the cost of methods to “fix” the cable(s) on the sea bed.
If a multiple turbine installation is to be considered, then this, also, must be considered. It is anticipated that, in these cases, a single large cable would be run to the centre of the turbine cluster and smaller cables run from a central point to the individual turbines. It has been suggested [4] that the total installed cost, for a 500 kW rated cable, might be around £40,000, for a 200 m cable, rising to £160,000 for an 1 km cable. These variations are shown in Fig. 8. The length of the cable is related to the relative locations of the turbine and the landfall location. It should be possible, a priori, to determine a function
Fig. 6. Possible variation in current speed with height above sea bed.

704 I. G. Bryden et al
Fig. 7. Simulated flow in the Berneray Sound, Outer Hebrides.
Fig. 8. Estimated cable costs for a 20 m diameter, 400 kW turbine.
for the cable-related costs in terms of cable length (L cable
) and required rating (R cable
). This function,
F cable
, must take into account cable costs per length, survey costs and installation costs.
3.2. Hardware costs
A feasibility study into tidal current generation in Orkney and Shetland [5] has identified certain key component costs for individual turbines. Some were related to physical size and some to electrical

Matching tidal current plants to local flow conditions
Turbine parameter
Table 2. Power rating and diameter of systems discussed for Orkney and Shetland [5].
Case 1 Case 2 Case 3
Rated power (kW)
Blade diameter (m)
97
10
217
15
386
20
705 size. The three cases studied in the report are outlined in Table 2. In each case, however, the cost breakdown was sufficiently detailed to allow separation of the influence of physical and electrical size.
The influence of rotor diameter on rotor costs is shown in Fig. 9. Similarly, the relationship between power rating and drivetrain cost is shown in Fig. 10. The power rating (P
R
) should be related to the maximum power according to the turbine power curve. In most models P
R will be the maximum power multiplied by an agreed factor. The manufacturer will be able to quote figures for the suggested size of this factor.
The cost of the support pillar is related to the water depth and, most critically, to the size (and mass) of the turbine blades. All of the component costs can be combined into a single function giving the device costs as a function, F(P
R
, Diam, Depth), of power rating (P
R
), rotor diameter (Diam) and water depth (Depth). It should be noted that this will not be a smooth function. It will rather be stepped, with the steps defined by manufacturing and supply factors.
Fig. 9. Estimated relationship between rotor diameter and rotor costs [5].
Fig. 10. Estimated relationship between power rating and drive train cost [5].

706 I. G. Bryden et al
3.3. Installation costs
Device installation costs were identified in the Orkney and Shetland feasibility study as being relatively independent of size and location within a channel and as such, do not feature directly in an optimisation exercise. For information, however, it is anticipated that they will be as shown in Table
3. This breakdown will, of course, become another influence if multiple installations are planned in which the mobilisation costs will not be repeated for each individual device.
4. DETERMINATION OF THE COST OF GENERATION
4.1. The cost model
In energy analysis, it is traditional to base the cost of energy generation, in this case electricity, on a discounted rate of return (DRR). The key factors which need to be included in the analysis are discussed in the following paragraphs.
4.1.1. Total fixed cost.
This represents the cost of the generation hardware, support structure, connection of the system to the grid and any other costs, such as those related to installation, which must be covered prior to the operation of the system.
4.1.2. Operation and maintenance costs (O and M).
This is frequently expressed as an annual rate proportional to the total capital cost. More complex models of O and M costs might be applied if they are available. Consideration of the maintenance schedule must involve a reduction in the effective load factor resulting from maintenance related down time. It is anticipated that a first-generation device might be subject to relatively frequent inspections but that major intervention would be much less regular.
4.1.3. Project lifetime.
Inevitably different elements of a tidal generation project will have differing lifetimes. The method of DRR can, however, be applied to different elements of the installation and to anticipated future maintenance schedule costs. Typically, however, it is common simply to specify a “payback period” over which the capital investment must be repaid. Once this period is over, the capital costs are assumed to be “written off” and its influence removed from consideration.
4.1.4. Discount rate.
The method of discounted rate of return has been suggested without criticism.
The question of what discount rate to assume is one which frequently causes debate; 5% used to be a frequently quoted figure for “public good” projects, although figures approaching 20% are not unheard of. In the Orkney and Shetland study, it was assumed that 8% represented a realistic figure for renewable energy projects.
4.1.5. Annual energy production.
This is, as has been outlined earlier, dependent upon the resource availability, the flow characteristics, the turbine design characteristics and turbine reliability.
4.2. Minimising costs with respect to the parametric description
Taking into account all of the factors influencing the costs of generation, it is possible to define a cost function of the form cost
= function(x i
, i
=
1, N), (8) where N is the number of design variables. Eq. (8) may be more conveniently expressed in vector notation by cost
=
f(x), (9)
Table 3. Estimated installation costs.
Activity
Mobilisation and transport
Drilling and pile installation
Turbine installation
Total
Cost
£150,000
£70,000
£30,000
£250,000

Matching tidal current plants to local flow conditions 707 where x is an N-dimensional vector representing the variable domain.
Some of these variables will be subject to constraints governed by other variables. The clearest example of this might be that the location of the turbine will have an influence upon the possible turbine blade diameter. These constraints may be expressed in the form g i
(x k
, k
=
1, N)
⬍ x i
⬍ h i
(x k
, k
=
1, N), (i
=
1, N), (10) where g(x)is a vector governing minimum constraints and h(x)is a vector governing maximum constraints.
The task for the designer is to find the values of the vector x which, subject to the specified constraints, will minimise the generation costs according to the model chosen as most appropriate. This is not a
Fig. 11. Sample input specification sheet for an optimisation process.
Fig. 12. Diagrammatic representation of look-up tables related to the gear box specification.

708 I. G. Bryden et al trivial exercise as, due to the nature of some of the dependencies, the problem is not linear. Indeed, mathematically some of the functions involved will not be continuous. For example, the costs of a generator will not be a smooth function of rating. It is much more likely that a manufacturer will have a range of generators which must be chosen. Many components of the turbine will, to save costs, be off the shelf. This also will result in discontinuous costing functions. It is inevitable that once the nature of the full costing function is identified, that it must be smoothed prior to optimisation otherwise the process is likely to be little better than a systematic variation in parameter value in a search for the optimal.
5. METHODOLOGY
The nature of the dependencies upon the state vector, x are such that it is unlikely that an analytic approach to optimisation can be established. Instead, an iterative approach should be adopted, probably with multiple random starting values to ensure that global minima rather than simply local minima are located. Initial approaches based upon the “solver” function in the spreadsheet Microsoft Excel have been encouraging, especially as this approach has allowed linkage with worksheets detailing the cost models. It is suggested that, if a spreadsheet aproach is used, separate sheets, such as that shown in
Fig. 11, are used for major components such as the gearbox.
In an example as shown, where the key variables relate to the availability of hardware from a manufacturer, care must be taken to ensure that the principal independent variables in a work sheet, in this case the “Opt-Current Reference Number”, although arbitrary in principle, represents a sensible pro-
Fig. 13. Samples from a tidal current parameterisation look-up table.

Matching tidal current plants to local flow conditions 709
Fig. 14. Sample final output sheet for an optimisation process.
gression from independent to dependent variables, such as maximum torque. The relationships between the reference number would be held in look-up tables, as shown diagramatically in Fig. 12. In the table the manufacturers’ data have been arranged so that there is a simple relationship between the independent variable, which in this case is a reference number, and the specification parameters, such as cost and maximum rotational speed, of the gearboxes.
In reality, prior to optimisation with respect to the independent variables, it may be necessary to replace the discrete values with continuous variables and functions. This could be achieved using a polynomial fit for the relationahips between independent and dependent variables. It will also be necessary to represent the output of the tidal simulation models in a parametric form. This will require the creation of a series of linked look-up tables, as shown in Fig. 13.
The key to the optimisation process depends upon the identification of the independent variables and the associated constraints. Once this is achieved, then iterative methods can be used to identify the values leading to an optimal solution. If a spreadsheet approach is used, then the iterative functions built into most modern spreadsheets can be utilised. Fig. 14 displays a possible model of the final page of a spreadsheet based optimisation system. It may, however, in the interests of user convenience, prove to be more convenient to develop dedicated software.
6. CONCLUSIONS
This paper outlines the procedures which might be utilised to achieve the cost-effective exploitation of the tidal current resource. It is not possible to generalise about just how big the levels of saving in using such methods actually are because they are so dependent upon local circumstances. What does appear likely is that, because early studies have indicated that the resource potential is large and that projected costs are close to marginal in comparison with fossil fuels, it would be unwise to attempt to develop the resource without serious attempts being made to identify optimal development procedures.
REFERENCES
1. Commission of the European Community (CEC), CENIX:—Tidal and Marine Currents Energy Exploitation
(JOU2-CT-93-0355), CEC-DGXVII (1995).
2. Energy Technology Support Unit, Tidal Stream Energy Review (ETSU T/05/00155/REP). DTI, UK (1993).
3. Bryden, I. G., Underwater Technology, 1994, 19(4), 7.
4. Bryden, I. G., Bullen, C. R., Baine, M. S. and Paish, O., Underwater Technology, 1995, 21(2), 21.
5. Bryden I. G. and Bullen C. R., Feasibility Study of Tidal Current Power Generation for Coastal Waters: Orkney and Shetland. Commission of the European Communities, CEC-DGXVII (1995).